Orthogonal polynomials: from Jacobi to Simon

69 Pages·2005·0.338 MB·English

by  Golinskii L., Totik V.

Orthogonal Polynomials: from Jacobi to Simon ∗
Leonid Golinskii and Vilmos Totik
June 29, 2005
Contents
1 Introduction 3
I General Theory 4
2 Orthogonal polynomials 5
Orthogonal polynomials with respect to measures . . . . . . . . . 5
The Riemann-Hilbert approach . . . . . . . . . . . . . . . . . . . 7
Orthogonal polynomials with respect to inner products . . . . . . 7
Varying weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Matrix orthogonal polynomials . . . . . . . . . . . . . . . . . . . 8
3 Classical orthogonal polynomials 8
4 Where do orthogonal polynomials come from? 10
Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Pad´e approximation and rational interpolation . . . . . . . . . . 11
Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Jacobi matrices and spectral theory of self-adjoint operators . . . 13
Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Some questions leading to classical orthogonal polynomials 15
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Polynomial solutions of eigenvalue problems . . . . . . . . . . . . 16
Harmonic analysis on spheres and balls . . . . . . . . . . . . . . 17
Approximation theory . . . . . . . . . . . . . . . . . . . . . . . . 17
∗SupportedbyINTAS03-51-6637(firstauthor)andbyNSFgrantDMS-040650andOTKA
T049448(secondauthor)
1
6 Heuristics 18
7 General orthogonal polynomials 20
8 Strong and ratio asymptotics 22
9 Exponential and Freud weights 26
10 Sobolev orthogonality 28
11 Non-Hermitian orthogonality 29
12 Multiple orthogonality 31
13 Matrix orthogonal polynomials 33
II Orthogonal polynomials on the unit circle 36
14 Definitions and basic properties 37
Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Szeg˝o recurrences and Verblunsky coefficients . . . . . . . . . . . 37
Bernstein – Szeg˝o approximation . . . . . . . . . . . . . . . . . . 39
15 Schur, Geronimus, Khrushchev 39
Schur functions and algorithm. . . . . . . . . . . . . . . . . . . . 39
Khrushchev’s theory . . . . . . . . . . . . . . . . . . . . . . . . . 40
16 Szeg˝o’s theory and extensions 41
17 CMV matrices 43
III Simon’s contribution 44
18 Analysis of CMV matrices 44
CMV matrices and spectral analysis . . . . . . . . . . . . . . . . 44
CMV matrices and the Szeg˝o function . . . . . . . . . . . . . . . 45
19 Zeros 46
Limit sets of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Mhaskar – Saff and clock theorems for zeros . . . . . . . . . . . . 47
2
20 Spectral theory in special classes 49
High order Szeg˝o theorem . . . . . . . . . . . . . . . . . . . . . . 49
Baxter’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B. Golinskii – I. Ibragimov condition . . . . . . . . . . . . . . . . 51
Sparse Verblunsky coefficients . . . . . . . . . . . . . . . . . . . . 51
Dense embedded point spectrum . . . . . . . . . . . . . . . . . . 52
Fibonacci subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . 52
21 Periodic Verblunsky coefficients 53
Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
22 Random coefficients 55
Decaying random Verblunsky coefficients. . . . . . . . . . . . . . 56
23 Miscellanea 57
Exponential decay estimates . . . . . . . . . . . . . . . . . . . . . 57
Rakhmanov’s theorem on an arc . . . . . . . . . . . . . . . . . . 58
Measures with VC’s from (cid:96)p, p>2 . . . . . . . . . . . . . . . . . 58
Counting eigenvalues in gaps . . . . . . . . . . . . . . . . . . . . 59
Jitomirskaya–Last inequalities . . . . . . . . . . . . . . . . . . . . 60
1 Introduction
Originally we were asked to write two separate papers. One on Barry Simon’s
work, and one on the state of the art in the theory of orthogonal polynomials.
However, Simon’s work on orthogonal polynomials is so fresh and fundamental
thatitwillbeforquitesomewhilethestateoftheartofthetheoryontheunit
circle. This conflict could have only resolved in a joint article.
This work is meant to non-experts, and therefore it contains some introduc-
torymaterial. Wetriedtolistmostoftheactivelyresearchedfields,butbecause
of space limitation we had one or two pages for areas where dozens of papers
and several books had been published. As a result our account is necessarily
incomplete.
The connection of orthogonal polynomials with other branches of math-
ematics is truly impressive, without even trying to be complete we mention
continued fractions, operator theory (Jacobi operators), moment problems, an-
alytic functions (Bieberbach’s conjecture), interpolation, Pad´e approximation,
quadrature, approximation theory, numerical analysis, electrostatics, statistical
quantum mechanics, special functions, number theory (irrationality and tran-
scendence),graphtheory(matchingnumbers),combinatorics,randommatrices,
stochasticprocesses(birthanddeathprocesses;predictiontheory),datasorting
and compression, Radon transform and computer tomography.
3
The theory of orthogonal polynomials can be divided into two main but
onlylooselyrelatedparts. Thetwopartshavemanythingsincommon,andthe
division line is quite blurred, it is more or less along algebra vs. analysis. One
of the parts is the algebraic aspect of the theory, which has close connections
with special functions, combinatorics and algebra, and it is mainly devoted
to concrete orthogonal systems or hierarchies of systems such as the Jacobi,
Hahn, Askey-Wilson, ... polynomials. All the discrete polynomials and the q-
analoguesofclassicalonesbelongtothistheory. Wewillnottreatthispart;the
interested reader can consult the three recent excellent monographs [39], [27]
and [4]. Much of the present state theory of orthogonal polynomials of several
variables lies also close to this algebraic part of the theory. To discuss them
would take us too far from our main direction, rather we refer the reader to the
recent book [23].
The other part is the analytical aspect of the theory. Its methods are ana-
lytical,anditdealswithquestionsthataretypicalinanalysis,orquestionsthat
have emerged in and related to other parts of mathematical analysis. General
properties fill a smaller part of the analytic theory, and the greater part falls
into two main and extremely rich branches: orthogonal polynomials on the real
line and on the circle. The richness is due to some special features of the real
line and the circle. Classical real orthogonal polynomials, sometimes in other
formslikecontinuedfractions,canbetracedbacktothe18thcentury,buttheir
rapid development occurred in the 19th and early 20th century. Orthogonal
polynomials on the unit circle is much younger, and their existence is largely
due to Szeg˝o and Geronimus in the first half of the 20th century. Simon’s re-
cent treatise [79, 80] (see also [90]) summarizes and greatly extends what has
happened since then.
The organization of the present article is as follows. First, in Part I we give
a brief outline of general and real orthogonal polynomials. Then we elaborate
on some recent trends and the state of the art of this branch of the analytic
theory. Simon’scontributionstorealorthogonalpolynomialswillbementioned
in this part. After that, in Part II, we move to orthogonal polynomials on the
circle, and, finally, Part III lists many of Simon’s contributions.
Eachofushasoptedforhisownstyleofexposition. PartI,preparedbythe
second author, deal mostly with the state of art in orthogonal polynomials and
covers areas/results that are from a period of over 100 years and from a large
number of people, therefore the style there is somewhat informal. In contrast,
Part III discusses mostly achievements of Barry Simon, and there more formal
statements are given.
4
Part I
General Theory
2 Orthogonal polynomials
Orthogonal polynomials with respect to measures
Let µ be a positive Borel measure on the complex plane with infinite support
for which (cid:90)
|z|mdµ(z)<∞
for all m>0. There are unique polynomials
p (z)=p (µ,z)=κ zn+···, κ >0, n=0,1,...
n n n n
which form an orthonormal system in L2(µ), i.e.
(cid:90) (cid:189)
0 if m(cid:54)=n
p p dµ=
m n 1 if m=n.
These p ’s are called the orthonormal polynomials corresponding to µ. κ is
n n
the leading coefficient, and p (z)/κ = zn+··· is called the monic orthogonal
n n
polynomial. The leading coefficients play a special and important role in the
theory, many properties depend on their behavior. When dµ(x) = w(x)dx on
some interval, say, then we talk about orthogonal polynomials with respect to
the weight function w.
The p ’s can be easily generated: all we have to do is to make sure that
n
(cid:90)
p (z)
n zkdµ(z)=0, k =0,1,...,n−1,
κ
n
whichisann×nsystemofequationsforthecoefficientsofp (z)/κ withmatrix
n n
(σ )n−1 , where
i,j i,j=0 (cid:90)
σ = zizjdµ(z)
i,j
are the complex moments of µ. This matrix is nonsingular, so the system has a
unique solution, and finally κ comes from normalization.
n
In particular, the complex moments already determine the polynomials. In
terms of them one can write up explicit determinant formulas:
(cid:175) (cid:175)
(cid:175) σ σ ··· σ 1 (cid:175)
(cid:175) 0,0 0,1 0,n−1 (cid:175)
(cid:175) σ σ ··· σ z (cid:175)
(cid:175) 1,0 1,1 1,n−1 (cid:175)
pn(z)= (cid:112) 1 (cid:175)(cid:175)(cid:175) ... ... ... ... ... (cid:175)(cid:175)(cid:175) (2.1)
Dn−1Dn (cid:175)(cid:175) σn−1,0 σn−1,1 ··· σn−1,n−1 zn−1 (cid:175)(cid:175)
(cid:175) (cid:175)
(cid:175) σ σ ··· σ zn (cid:175)
n,0 n,1 n,n−1
5
where
D =|σ |n (2.2)
n i,j i,j=0
are the so called Gram determinants.
Note that if µ is supported on the real line then
(cid:90)
σ = xi+jdµ(x)=:α ,
i,j i+j
soD =|α |n isaHankeldeterminant, whileifµissupportedontheunit
n i+j i,j=0
circle then (cid:90)
σ = zi−jdµ(z)=:β ,
i,j i−j
so D =|β |n is a Toeplitz determinant. In these two important cases the
n i−j i,j=0
orthogonal polynomials have many special properties that are missing in the
general theory. For example, in the real case, i.e., if µ is supported on the real
line, the p ’s obey a three term recurrence formula
n
xp (x)=a p (x)+b p (x)+a p (x), (2.3)
n n n+1 n n n−1 n−1
where (cid:90)
κ
a = n >0, b = xp2(x)dµ(x),
n κ n n
n+1
and conversely, any system of polynomials satisfying (2.3) with real a >0, b
n n
isanorthonormalsystemwithrespecttoa(notnecessarilyunique)measureon
the real line (Favard’ theorem). In the real case the zeros of p are real and
n
simple and the zeros of p and p interlace, i.e. in between any two zeros of
n n+1
p there is a zero of p . We emphasize that the three term recurrence is a
n+1 n
very special property of real orthogonal polynomials, and it is due to the fact
that in this case the polynomials are real, hence
(cid:90) (cid:90)
xp (x)p (x)dµ(x)= p (x)(xp (x))dµ(x)=0
n m n m
form<n−1. Thisthreetermrecurrenceismissinginthegeneralcase,anditis
replacedbyadifferentrecurrenceforpolynomialsonthecircle(seePartII).For
example, in the real case the three term recurrence implies for the reproducing
kernel the Christoffel-Darboux formula
(cid:88)n κ p (x)p (t)−p (x)p (t)
p (x)p (t)= n n+1 n n n+1 .
k k κ x−t
n+1
k=0
The starting values of the recurrence (2.3) are p ≡ 0, p = (µ(C)))−1/2.
−1 0
Ifonestartsfromq =−1,q ≡0andusethesamerecurrence(witha =1)
−1 0 −1
xq (x)=a q (x)+b q (x)+a q (x), (2.4)
n n n+1 n n n−1 n−1
6
thenq isofdegreen−1,andbyFavard’stheoremthedifferentq ’sareorthog-
n n
onal with respect to some measure. The q ’s are called orthogonal polynomials
n
of the second kind (sometimes for p we say that they are of the first kind).
n
They can also be written in the form
(cid:90)
p (z)−p (x)
q (z)=(µ(C))−1/2 n n dµ(x).
n z−x
The Riemann-Hilbert approach
Let µ still be supported on the real line, and suppose that it is of the form
dµ(t) = w(t)dt with some smooth function w. A new approach to generating
orthogonalpolynomialsthathasturnedouttobeofgreatimportancewasgiven
in the early 1990’s by Fokas, Its and Kitaev [26]. Consider 2×2 matrices
(cid:181) (cid:182)
Y (z) Y (z)
Y(z)= 11 12
Y (z) Y (z)
21 22
where the Y are analytic functions on C\R, and solve for such matrices the
ij
following matrix valued Riemann-Hilbert problem:
1. for all x∈R (cid:181) (cid:182)
1 m(x)
Y (x)=Y (x)
+ − 0 1
where Y resp. Y is the limit of Y(z) as z tends to x from the upper resp.
+ −
lower half plane, and
2. (cid:181) (cid:181) (cid:182)(cid:182)(cid:181) (cid:182)
1 zn 0
Y(z)= I+O
z 0 z−n
at infinity, where I denotes the identity matrix.
There is a unique solution Y(z), and its entry Y (z) is precisely the monic
11
polynomial p (µ,z)/κ . The other entries can also be explicitly written in
n n
terms of p and p , furthermore κ and the recurrence coefficients a ,b can
n n−1 n n n
be expressed from the entries of Y , where Y is the matrix in
1 1
(cid:181) (cid:182) (cid:181) (cid:182)
z−n 0 1 1
Y(z) =I +Y +O .
0 zn 1z z2
FordetailsonthisRiemann-Hilbertapproachsee[19]or[18]inthisvolume.
Orthogonal polynomials with respect to inner products
Sometimes one talks about orthogonal polynomials with respect to an inner
product(cid:104)·,·(cid:105)whichisdefinedonsomelinearspacecontainingallpolynomials. In
this case orthogonality means (cid:104)p ,p (cid:105)=0 for m(cid:54)=n. When the inner product
n m
is positive definite in the sense that (cid:104)p,p(cid:105) = 0 only for the zero polynomial
7
p, then the aforementioned orthogonalization process can be used, and with
σ =(cid:104)xi,xj(cid:105), the determinantal formula (2.1) is still valid. The same is true if
i,j
the Gram determinants (2.2) are different from zero. However, if this is not so,
e.g. in the so called non-Hermitian orthogonality (see section 11), then these
cannot be used. In this case we write
p (z)=γ zn+γ zn−1+···,
n n n−1
and make sure that p is orthogonal to all powers zk, 0≤k <n, i.e., solve the
n
homogeneous system of equations
(cid:88)n
γ σ =0, k =0,...,n−1
j j,k
j=0
for γ ,γ ,...,γ . Since the number of unknowns is bigger than the number of
0 1 n
equations, there is always a non-trivial solution, which gives rise to non-trivial
orthogonal polynomials. However, we cannot assert any more γ (cid:54)= 0, so the
n
degree of p may be smaller than n, and there may be several choices for p .
n n
Still, in applications where non-Hermitian orthogonality is used, these p play
n
the role of orthogonal polynomials.
Varying weights
In the last 25 years orthogonal polynomials with respect to varying measures
have played significant role in several problems, see e.g. the sections on expo-
nentialandFreudweightsoronrandommatricesinSection4. Informingthem
one has a sequence of measures µ (generally with some particular behavior),
n
and for each n one forms the orthogonal system {p (µ ,z)}∞ . In most cases
k n k=0
one needs the behavior of p (µ ,z) or that of p (µ ,z) with some fixed k.
n n n±k n
Matrix orthogonal polynomials
Orthogonalityofmatrixpolynomials(i.e. whentheentriesofthefixedsizema-
trix are polynomials of degree n=0,1,... and orthogonality is with respect to
amatrixmeasure)isaveryactiveareawhichshowsextremerichnesscompared
to the scalar case. See section 13 for a short discussion.
3 Classical orthogonal polynomials
These are
• Jacobi polynomials P(α,β), α,β > −1, orthogonal with respect to the
n
weight (1−x)α(1+x)β on [−1,1],
8
• Laguerre polynomials L(α), α >−1, with orthogonality weight xαe−x on
n
[0,∞),
• andHermitepolynomialsH orthogonalwithrespecttoe−x2 on(−∞,∞).
n
In the literature various normalizations are used for them.
They are very special, for they possess many properties that no other or-
thogonal polynomial system does. In particular,
• they have derivatives which form again an orthogonal polynomial system,
e.g. the derivative of P(α,β) is a constant multiple of P(α+1,β+1):
n n−1
1
(P(α,β))(cid:48)(x)= (n+α+β+1)P(α+1,β+1)(x),
n 2 n−1
• they all possess a Rodrigue’s type formula
1 dn
P (x)= {w(x)σ(x)n},
n d w(x)dxn
n
where w is the weight function and σ is a polynomial that is independent
of n, for example,
1 dn (cid:161) (cid:162)
L(α)(x)=exx−α e−xxn+α ,
n n!dxn
• they satisfy a differential-difference relation of the form
π(x)P(cid:48)(x)=(α x+β )P (x)+γ P (x),
n n n n n n−1
e.g.
x(L(α))(cid:48)(x)=nL(α)(x)−(n+α)L(α) (x),
n n n−1
• they satisfy a non-linear equation of the form
σ(x)(P (x)P (x))(cid:48) =(α x+β )P (x)P (x)+γ P2(x)+δ P2 (x),
n n−1 n n n n−1 n n n n−1
with some constants α ,β ,γ ,δ , and σ a polynomial of degree at most
n n n n
2, e.g.
(H (x)H (x))(cid:48) =2xH (x)H (x)−H2(x)+2nH2 (x).
n n−1 n n−1 n n−1
Noweveryoneofthesehasaconverse,namelyifasystemoforthogonalpolyno-
mials possesses any of these properties, then it is (up to a change of variables)
one of the classical systems [1]. See also Bochner’s result in the next section
claimingthattheclassicalorthogonalpolynomialsareessentiallytheonlypoly-
nomial (not just orthogonal polynomial) systems that satisfy a certain second
order differential equation.
Classical orthogonal polynomials are also special in the sense that they pos-
sess a relatively simple
9
• second order differential equation, e.g.
xy(cid:48)(cid:48)+(α+1−x)y(cid:48)+ny =0
for L(α),
n
• generating function, e.g.
(cid:88)H (x)
n wn =exp(2xw−w2),
n!
n
• integral representation, e.g.
(cid:90)
(−1)n
(1−x)α(1+x)βP(α,β)(x)= (1−t)n+α(1+t)n+β(t−x)−n−1dt
n 2n+1πi
over an appropriate contour,
and these are powerful tools to study their behavior.
For all these results see [104].
4 Where do orthogonal polynomials come from?
In this section we mention a few selected areas where orthogonal polynomials
naturally arise.
Continued fractions
Continuedfractionsplayedextremelyimportantroleinthedevelopmentofsev-
eral branches of mathematics, but their significance has been unjustly dimin-
ished in modern mathematics. A continued fraction is of the form
B
1 ,
A1+ A2B+2···
and its n-th convergent is
S B
n = 1 , n=1,2,....
Rn A1+ A2+B··2·BAnn
The value of the continued fraction is the limit of its convergents. The de-
nominatorsandnumeratorsoftheconvergentssatisfythethreetermrecurrence
relations
R = A R +B R , R ≡1, R ≡0
n n n−1 n n−2 0 −1
S = A S +B S , S ≡0, S ≡1,
n n n−1 n n−2 0 −1
10